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  "body": "\n## Definition\n\nLet $A$ be a unital C*-algebra with unit $1$ and involution $a \\mapsto a^*$.\nAn *operator system* in $A$ is a linear subspace $S \\subseteq A$ such that:\n\n1. $1 \\in S$;\n2. $a^* \\in S$ for every $a \\in S$.\n\n## Remarks\n\nExample: in the C*-algebra $C(\\mathbb{T})$ of continuous functions on the unit circle $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$, let $S$ be the span of $1, z, \\bar z$.\nIt contains $1$, and $(a + b z + c \\bar z)^* = \\bar a + \\bar c z + \\bar b \\bar z$ lies in $S$, so $S$ is an operator system.\nNon-example: the span of $1$ and $z$ in $C(\\mathbb{T})$ contains $1$ but not $\\bar z$.\nIndeed, if $\\bar z = a + b z$ on $\\mathbb{T}$, then $z = 1$ and $z = -1$ give $a = 0$ and $b = 1$, but then $z = i$ gives $-i = i$.\nEvery self-adjoint element $h$ of an operator system $S$ satisfies $-\\norm{h} 1 \\le h \\le \\norm{h} 1$, so $S$ is spanned by its positive elements.\nMany authors take $A = B(H)$ for a Hilbert space $H$, and some also require $S$ to be norm closed.\n",
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